### Existence and uniqueness of positive solutions to certain differential systems

#### Abstract

In this article we study existence and uniqueness of positive solutions for elliptic systems of the form \begin{align} -\Delta v = &f(x,u) \quad \hbox{in} \quad \Omega \\ -\Delta u = & v^\beta \quad \hbox{in}\quad \Omega, \end{align} with Dirichlet boundary condition on a bounded smooth domain in $\Bbb R^N$. The nonlinearity $f$ is assumed to have a sub-$\beta$ growth with $\beta>0$, that in case $f(x,u)=u^\alpha, \alpha>0$, corresponds to $\alpha\beta<1$. The results are also valid for a larger class of systems, including some infinite dimensional Hamiltonian Systems.

#### Article information

Source
Adv. Differential Equations, Volume 3, Number 4 (1998), 575-593.

Dates
First available in Project Euclid: 18 April 2013

Mathematical Reviews number (MathSciNet)
MR1659242

Zentralblatt MATH identifier
0946.35028

#### Citation

Felmer, Patricio; Martínez, Salomé. Existence and uniqueness of positive solutions to certain differential systems. Adv. Differential Equations 3 (1998), no. 4, 575--593. https://projecteuclid.org/euclid.ade/1366292564